On Arboreal Galois Representations of Rational Functions

The action of the absolute Galois group $\text{Gal}(K^{\text{ksep}}\mathrm{/}K)$ of a global field $K$ on a tree $T(\varphi ,\alpha )$ of iterated preimages of $\alpha \in $\mathbb{P}$^1(K)$under$\phi \in K(x)$with$$\text{deg}$(\phi) \geq 2$inducesahomomorphism$\rho: $\text{Gal}$(K^{$\text{ksep}$}/K) \to $\text{Aut}$(T(\phi, \alpha))$, which is called an arboreal Galois representation. In this paper, we address a number of questions posed by Jones and Manes about the size of the group $G(\phi,\alpha) := $\text{im}$ \rho = \underset{\leftarrow n}\lim$\text{Gal}$(K(\phi^{-n}(\alpha))/K)$. Specifically, we consider two cases for the pair $(\varphi ,\alpha )$: (1) $\varphi$ is such that the sequence $\{a_n\}$ defined by $a_0=\alpha$ and $a_n=\varphi (a_{n-1})$ is periodic, and (2) $\varphi$ commutes with a nontrivial Mobius transformation that fixes $\alpha$. In the first case, we resolve a question posed by Jones about the size of $G(\varphi ,\alpha )$, and taking $K=\mathbb{Q}$, we describe the Galois groups of iterates of polynomials $\varphi \in \mathbb{Z}[x]$ that have the form $\phi(x) = x^2 + kx$or$\phi(x) = x^2 - (k+1)x + k$\mathrm{.}When$K = $\mathbb{Q}$$and$\phi \in $\mathbb{Z}$[x]$, arboreal Galois representations are a useful tool for studying the arithmetic dynamics of $\varphi$. In the case of $\varphi (x)=x^2+kx$ for $k\in \mathbb{Z}$, we employ a result of Jones regarding the size of the group $G(\psi ,0)$, where $\psi (x)=x^2-kx+k$, to obtain a zero-density result for primes dividing terms of the sequence $\{a_n\}$ defined by $a_0 \in $\mathbb{Z}$$and$a_n = \phi(a_{n-1})$. In the second case, we resolve a conjecture of Jones about the size of a certain subgroup $C(\varphi ,\alpha )\subset \text{Aut}(T(\varphi ,\alpha ))$ that contains $G(\varphi ,\alpha )$, and we present progress toward the proof of a conjecture of Jones and Manes concerning the size of $G(\varphi ,\alpha )$ as a subgroup of $C(\varphi ,\alpha )$.